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The universal $R$-matrix of a quantized universal enveloping algebra is typically written as the product of two terms, one only involving elements of the Cartan, and one only involving elements of the upper and lower triangular pieces of $U_q(g)$.

How do I think about commuting these individual pieces past things like coproducts? Are calculations like this written somewhere?

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up vote 3 down vote accepted

Another way to think about this, which I learned from Mark Haiman's class, is that there are actually $4$ natural coproducts on the quantum group. $\Delta\left(E\right)$ could be $E\otimes K + 1 \otimes E$, or you could move the $K$ to the other factor, or you could invert $K$, or you could do both. The individual parts of the $R$-matrix move between these four different coproducts. In particular, the quasi-$R$-matrix (iirc) turns the $K$ into a $K^{-1}$.

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The answer is in Lusztig's preposterously short PNAS paper on canonical bases for tensor products. He calls this element the "quasi-R-matrix."

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